Theorem subrginv 18796

Description: A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrginv.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrginv.2	⊢ 𝐼 = (invr‘𝑅)
subrginv.3	⊢ 𝑈 = (Unit‘𝑆)
subrginv.4	⊢ 𝐽 = (invr‘𝑆)

Assertion

Ref	Expression
subrginv	⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Proof of Theorem subrginv

Step	Hyp	Ref	Expression
1		subrgrcl 18785	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2	1	adantr 481	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring)
3		subrginv.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
4	3	subrgbas 18789	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	subrgss 18781	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
7	4, 6	eqsstr3d 3640	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
8	7	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
9	3	subrgring 18783	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		subrginv.3	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑆)
11		subrginv.4	. . . . . . 7 ⊢ 𝐽 = (invr‘𝑆)
12		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	10, 11, 12	ringinvcl 18676	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
14	9, 13	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
15	8, 14	sseldd 3604	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑅))
16	12, 10	unitcl 18659	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑆))
17	16	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
18	8, 17	sseldd 3604	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
19		eqid 2622	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	3, 19, 10	subrguss 18795	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
21	20	sselda 3603	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Unit‘𝑅))
22		subrginv.2	. . . . . . 7 ⊢ 𝐼 = (invr‘𝑅)
23	19, 22, 5	ringinvcl 18676	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Base‘𝑅))
24	1, 23	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Base‘𝑅))
25	21, 24	syldan 487	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ (Base‘𝑅))
26		eqid 2622	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
27	5, 26	ringass 18564	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝐽‘𝑋) ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝐼‘𝑋) ∈ (Base‘𝑅))) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))))
28	2, 15, 18, 25, 27	syl13anc 1328	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))))
29		eqid 2622	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
30		eqid 2622	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
31	10, 11, 29, 30	unitlinv 18677	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
32	9, 31	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
33	3, 26	ressmulr 16006	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
34	33	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆))
35	34	oveqd 6667	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = ((𝐽‘𝑋)(.r‘𝑆)𝑋))
36		eqid 2622	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	3, 36	subrg1 18790	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
38	37	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (1r‘𝑅) = (1r‘𝑆))
39	32, 35, 38	3eqtr4d 2666	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
40	39	oveq1d 6665	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)))
41	19, 22, 26, 36	unitrinv 18678	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
42	1, 41	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
43	21, 42	syldan 487	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
44	43	oveq2d 6666	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
45	28, 40, 44	3eqtr3d 2664	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
46	5, 26, 36	ringlidm 18571	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑋) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
47	2, 25, 46	syl2anc 693	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
48	5, 26, 36	ringridm 18572	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐽‘𝑋) ∈ (Base‘𝑅)) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
49	2, 15, 48	syl2anc 693	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
50	45, 47, 49	3eqtr3d 2664	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Basecbs 15857   ↾_s cress 15858  ._rcmulr 15942  1_rcur 18501  Ringcrg 18547  Unitcui 18639  inv_rcinvr 18671  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-subrg 18778

This theorem is referenced by:  subrgdv  18797  subrgunit  18798  subrgugrp  18799  issubdrg  18805  gzrngunit  19812